If you use extensions (Firefox/Tools > Add-ons > Extensions) that can block content (e.g. Adblock Plus, NoScript, Flash Block, Ghostery) then make sure that such extensions aren't blocking content.

You can remove all data stored in Firefox from a specific domain via "Forget About This Site" in the right-click context menu of an history entry ("History > Show All History" or "View > Sidebar > History") or via the about:permissions page.

Using "Forget About This Site" will remove all data stored in Firefox from that domain like bookmarks, cookies, passwords, cache, history, and exceptions, so be cautious and if you have a password or other data from that domain that you do not want to lose then make a note of those passwords and bookmarks.

You can't recover from this 'forget' unless you have a backup of the involved files.

It doesn't have any lasting effect, so if you revisit such a 'forgotten' website then data from that website will be saved once again.

You can check for problems with current Flash plugin versions and try these:

disable a possible RealPlayer Browser Record Plugin extension for Firefox and update the RealPlayer if installed

I'm seeing a hidden Flash player on bandcamp.com playing the sound and disabling Flash makes the sound stop for me.
There are only audio tags in the page code, but they are empty (i.e. no source specified).

Okay. I think I found out that there is no audio on Firefox on any HTML5 player. I tried the one at home page of http://mediaelementjs.com/ it shows it is playing but there is no audio (there is video though). And below video player there is an audio player too - no sound.

I cleared all caches and settings and disabled all addons and extensions and updated to all latest flashes and everything.

You can click the Volume icon on the Windows Taskbar near the clock and click the Mixer link at the bottom to open the mixer.
You can adjust the volume level for each application in that window, so make sure that Firefox is not muted or set at a low level.
Current Firefox versions use the plugin-container process to run plugins, so you need to check the volume setting for the plugin-container process.
You will only see the Firefox program or the Plugin Container in the Mixer if you are on a website that plays sound.

Yeah for me it also says that my browser doesn't have MPEG-4 support, but at http://caniuse.com/mpeg4 it says that latest FF has support for it. So I don't get it at all.

Also for anyones interest since I started this thread, soundcloud was working fine, but maybe couple days ago it also started not to work, it plays, but there is no sound. Just like that, for no reason. I even reset all my Firefox settings, neither safe mode with all addons disabled or this, do not help.

Seems that those changing formats (Flash, HTML5, and so on) are way more picky on system/OS configurations.
For me http://caniuse.com/mpeg4 says that FF 23 is partially supported.
And Soundcloud is giving no troubles at all, can hear all the clips.
And YouTube with HTML5 there enabled is around 50/50 for me when it comes to audio with the videos.
Here on Windows 8 Pro 64bit.

Finally, after hundreds of hours I fixed it and Firefox plays sound in HTML5 in everywhere, YouTube (HTML5 mode), SoundCloud, Bandcamp and etc.

You have to: Right-click on your sound/speakers icon in the system tray and select Playback Devices, go to Speakers properties (the one which is green-ticked, also disable the red ones, if any), then go to Enhancements and tick Disable all sound effects. And that's it, restart the browser and enjoy.

Also, if didn't help, make sure you have media.windows-media-foundation.enabled set to 'true' in about:config.

You should also have some decent codecs installed in your machine, I have VLC Player and it comes with lots of in-built codecs. Also make sure you have VLC Web Plugin enabled in Add-ons -> Extensions (Always activate)

So the unanswered is who's problem is that? Is it Firefox? Or Windows 8? Or my Realtek audio drivers fault? (Anyway without disabling sound effects, HTML5 works fine in any other browser, so I blame Firefox for this nonsense for extra configurations in Windows audio settings)

That indeed works! :)
But, think it's indeed more a treatment of a symptom with Firefox then the solution to the problem. The problem itself still is there :(
Besides that: the audio is there now but now the video(s) in HTML5 are jerky, freezing for a second every 5 seconds orso (at YouTube) while the video allready is completely loaded.
But at least there is a kind of workaround for this problem now :)

Hmm, I don't know. I don't have these issues now at all. Maybe you are trying to watch HD content on older machine? Or maybe slow internet connection? I am watching 1080p HD HTML5 videos on YouTube now with i3 computer and 100Mbps optic fiber line.

Anyway, could be anything, and yes, this problem should be fixed by Firefox, someone should report somewhere.

I have the same problem, HTML5 video does not play sound to my computer speaker but instead to my TV speakers!

Tried disabling my LG TV audio in playback device and also disable all sound effect but the sound still output via ARC through HDMI. My default onboard sound is the default audio playback for all formats. My mediaplayer play without problem, youtube play flash to my speakers, only HTML5 plays the audio to my LG TV speakers!

Is there any away to configure the playback device for HTML5 audio?????

You can try to set the media.windows-media-foundation.enabled pref to false on the about:config page to disable the built-in mp4 media player.

Bandcamp works after I do that. However, lots of other built-in media still doesn't play (such as the drop on Relevantmag.com) in Firefox. I'm really losing faith in Firefox and don't want to switch to chrome, but I may have to :(